1. Y-axis
2. Origin
3. X-axis, Y-axis and Origin
4. X-axis, Y-axis and Origin
5. X-axis
6. X-axis
7. X-axis, Y-axis and Origin
8. \((x – 2)^2 + (y + 1)^2 = 25\)
9. \((x + 3)^2 + (y – 2)^2 = 5\)
10. \(x^2 + y^2 = 25\)
11. \((x – 1)^2 + (y + 1)^2 = 50\)
12. \((x – 1)^2 + (y + 3)^2 = 9\)
13. \((x + 4)^2 + (y – 1)^2 = 16\)
14. \((x – 3)^2 + (y – 1)^2 = 10\)
15. \((x – 1)^2 + y^2 = 1\)
16. \((x – 3)^2 + (y – 4)^2 = 25\)
17. Center \((1, \, 0), \; r= 2\)
18. Center \((0, \, -2), \; r = 2 \sqrt{2}\)
19. Center \(\left( 0, \, -\frac{1}{2} \right), \; r = \frac{1}{2} \)
20. Center \((1, \, -2), \; r = 3 \)
21. Center \(\left( \frac{1}{4}, \, -\frac{1}{4} \right), \; r = \frac {\sqrt{10}}{4}\)
22. Center \(\left( \frac{3}{2}, \, \frac{1}{2} \right), \; r = \frac{9}{4}\)
23. \[ (y – 1)^2 = (x + 1)^3 \]

    

24. \[ (x – 1)^2 = (y + 1)^3 \]

    

25. \[ (y + 1)^2 = (x – 1)^3 \]

    

26. \[ (x-3)^2 (y – 2) = 4 \left(2 -(y – 2) \right) \]

    

27. \[ (y – 3)^2 (x – 2)= 4 \left( 2 – (x-2) \right) \]

    

28. \[ (x + 3)^2 (y + 2) = 4 \left( 2 -(y + 2) \right) \]

    

In exercises 1 through 7, use the test of symmetry to determine if the graph of the equation is symmetric with respect to the X-axis, Y-axis, or the origin.

1. \[ y = x^2 \]
2. \[ xy = 1 \]
3. \[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
4. \[ \frac{x^2}{4} – \frac{y^2}{9} = 1 \]
5. \[ y^2(2 – x) = x^3 \]
6. \[ x^2 + y^2 + x = \sqrt{x^2+y^2} \]
7. \[ (x^2 + y^2)^2 = x^2 – y^2 \]

In exercises 8 through 16, find an equation of the circle that satisfies the given conditions.

8. Center, \((2, -1)\);   \(r = 5\).
9. Center \((-3, 2)\);   \(r =\sqrt{5}\).
10. Center in the origin, pass through \((-3, 4)\).
11. Center \((1, -1)\), pasa por \((6, 4)\).
12. Center \((1, -3)\), es tangente al eje X.
13. Center \((-4, 1)\), es tangente al eje Y.
14. A diameter with endpoints: \((2, 4)\)   and   \((4, -2)\).
15. Radius \(r = 1\) pass through: \((1, 1)\)   and   \((1, -1)\).
16. Passing through the points \((0, 0)\), \((0, 8)\)   and   \((6, 0)\).

In exercises 17 through 22, prove that the equation corresponds to a circle by finding its center and radius.

17. \[ x^2 + y^2 – 2x – 3 = 0 \]
18. \[ x^2 + y^2 + 4y – 4 = 0 \]
19. \[ x^2 + y^2 + y = 0 \]
20. \[ x^2 + y^2 – 2x + 4y – 4 = 0 \]
21. \[ 2x^2 + 2y^2 – x + y – 1 = 0 \]
22. \[ 16x^2 + 16y^2 – 48x – 16y – 41 = 0 \]

In exercises 23, 24, and 25, graph the equations by applying the translation criterion to the semi-cubical parabola (ex. 2.2.7-b).

23. \[ (y – 1)^2 = (x + 1)^3 \]
24. \[ (x – 1)^2 = (y + 1)^3 \]
25. \[ (y+1)^2 = (x – 1)^3 \]

In exercises 26 through 28, graph the equation by applying the translation and inversion criteria to the graph of the Witch of Agnesi (example 2.2.7-a).

26. \[ (x-3)^2(y-2)=4(4-y) \]
27. \[ (y – 3)^2(x – 2) = 4(4 – x) \]
28. \[ (x + 3)^2(y + 2) = 4(-y) \]